MathTrek

In the realm of counted cross stitch, Mary D. Shepherd of Northwest Missouri State University displayed her painstakingly woven symmetry patterns. For the type of cloth and technique that she uses, the fabric is a grid of squares, and one cross stitch covers one square of the fabric. The only possible subdivision of this square is with a stitch that "covers" half a square on the diagonal, Shepherd says.

These features constrain the number of symmetry patterns that you can weave. Of 17 possible wallpaper patterns, for example, only 12 can be done in counted cross stitch.

Six of the 12 wallpaper patterns that can be done in counted cross stitch needlework. Courtesy of Mary D. Shepherd

Shepherd has also worked on frieze and rosette symmetry patterns. Rosette patterns, for example, give a nice visualization of the symmetries of a square (technically, the group D4 and all its subgroups), she says.

Rosette patterns for visualizing the symmetries of a square (the dihedral group of the square). Courtesy of Mary D. Shepherd

Jake Wildstrom is a graduate student at the University of California, San Diego. His passion is crocheting in relief.

One of the few fractals that's amenable to crochet is the Sierpinski triangle. Wildstrom has turned this remarkable geometric figure into blankets, wispy shawls, and even a hat. His instructions for crocheting these figures can be found at http://www.math.ucsd.edu/%7Edwildstr/crochet/sierpinski.html.

Jake Wildstrom's relief crocheting has turned a fractal known as the Sierpinksi triangle into a shawl. Photo by I. Peterson.

Mathematical origami design was well represented by Tom Hull of Merrimack College (see http://www.merrimack.edu/~thull/).

One of Tom Hull's modular origami creations. Photo by I. Peterson.

Laura M. Shea of Parker, Colo., strings tiny crystal beads to form polyhedra or geometric tilings.

This beadwork bracelet, created by Laura Shea, is based on a triangular tiling. Photo by I. Peterson.

Creating polyhedra with beads is an interesting way to learn the properties of regular and semi-regular solids, Shea says. In a bead polyhedron, each face becomes open space, each edge becomes one bead, and each vertex becomes a thread void. The resulting structure is light and open.

The "Plato Bead," created by Laura Shea, is a dodecahedron. A bead stands in for each of this polyhedron's 30 edges. Each of the 20 vertices becomes a void surrounded by three beads and thread. The 12 faces of the form become open spaces. Courtesy of Laura Shea.

Shea is now working on a book about geometrically inspired beadwork.

No mathematical crafts session of the knitting network would be complete without a Möbius band—that mind-bending, one-sided, one-edged object. Josh Holden of the Rose-Hulman Institute of Technology spent his time crocheting one.

Josh Holden's crocheted Möbius band. Photo by I. Peterson.

And there were even a few people actually knitting at the knitting network session.

Carolyn Yackel works on her knitting. Photo by I. Peterson.

belcastro, Yackel, and a number of other crafters are preparing a book that will contain not only instructions for creating mathematical objects but also insights into the underlying mathematics. A K Peters will publish the book later this year.

The 2005 JMM featured a special session of presentations on mathematics and mathematics education in fiber arts. For more information on that session, see http://www.toroidalsnark.net/mkss.html. Photos of that year's fiber arts exhibit and knitting network event are available at http://www.toroidalsnark.net/mkexh2005/mkexh2005.html and http://community.livejournal.com/mathart/7266.html.

References:

belcastro, s.-m., and Carolyn Yackel. 2006. About knitting . . . . Math Horizons (November):24-27.

Klarreich, E. 2006. Crafty geometry. Science News 170(Dec. 23&30):411-413. Available at http://www.sciencenews.org/articles/20061223/bob10.asp.

Mary D. Shepherd has a Web page, with a link to a paper about counted cross stitch and symmetry, at http://catpages.nwmissouri.edu/m/msheprd/shepherd.htm.

Jake Wildstrom's home page is at http://www.math.ucsd.edu/%7Edwildstr/, with material on crocheting mathematical figures at http://www.math.ucsd.edu/%7Edwildstr/crochet/.

Tom Hull's Web page is at http://www.merrimack.edu/~thull/. His origami math pages are available at http://www.merrimack.edu/~thull/origamimath.html.

To see Laura M. Shea's beading Web site, go to http://www.adancingrainbow.com/. For additional examples of her work, go to http://www.bridgesmathart.org/art-exhibits/jmm07/shea.html.

sarah-marie belcastro's Web site at http://www.toroidalsnark.net/ provides lots of material on mathematical knitting (http://www.toroidalsnark.net/mathknit.html) and other pastimes.